3.46 \(\int \cot ^9(c+d x) (a+a \sec (c+d x))^3 \, dx\)

Optimal. Leaf size=149 \[ \frac {9 a^3}{4 d (1-\cos (c+d x))}+\frac {a^3}{32 d (\cos (c+d x)+1)}-\frac {39 a^3}{32 d (1-\cos (c+d x))^2}+\frac {5 a^3}{12 d (1-\cos (c+d x))^3}-\frac {a^3}{16 d (1-\cos (c+d x))^4}+\frac {57 a^3 \log (1-\cos (c+d x))}{64 d}+\frac {7 a^3 \log (\cos (c+d x)+1)}{64 d} \]

[Out]

-1/16*a^3/d/(1-cos(d*x+c))^4+5/12*a^3/d/(1-cos(d*x+c))^3-39/32*a^3/d/(1-cos(d*x+c))^2+9/4*a^3/d/(1-cos(d*x+c))
+1/32*a^3/d/(1+cos(d*x+c))+57/64*a^3*ln(1-cos(d*x+c))/d+7/64*a^3*ln(1+cos(d*x+c))/d

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Rubi [A]  time = 0.10, antiderivative size = 149, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {3879, 88} \[ \frac {9 a^3}{4 d (1-\cos (c+d x))}+\frac {a^3}{32 d (\cos (c+d x)+1)}-\frac {39 a^3}{32 d (1-\cos (c+d x))^2}+\frac {5 a^3}{12 d (1-\cos (c+d x))^3}-\frac {a^3}{16 d (1-\cos (c+d x))^4}+\frac {57 a^3 \log (1-\cos (c+d x))}{64 d}+\frac {7 a^3 \log (\cos (c+d x)+1)}{64 d} \]

Antiderivative was successfully verified.

[In]

Int[Cot[c + d*x]^9*(a + a*Sec[c + d*x])^3,x]

[Out]

-a^3/(16*d*(1 - Cos[c + d*x])^4) + (5*a^3)/(12*d*(1 - Cos[c + d*x])^3) - (39*a^3)/(32*d*(1 - Cos[c + d*x])^2)
+ (9*a^3)/(4*d*(1 - Cos[c + d*x])) + a^3/(32*d*(1 + Cos[c + d*x])) + (57*a^3*Log[1 - Cos[c + d*x]])/(64*d) + (
7*a^3*Log[1 + Cos[c + d*x]])/(64*d)

Rule 88

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

Rule 3879

Int[cot[(c_.) + (d_.)*(x_)]^(m_.)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n_.), x_Symbol] :> Dist[1/(a^(m - n
- 1)*b^n*d), Subst[Int[((a - b*x)^((m - 1)/2)*(a + b*x)^((m - 1)/2 + n))/x^(m + n), x], x, Sin[c + d*x]], x] /
; FreeQ[{a, b, c, d}, x] && IntegerQ[(m - 1)/2] && EqQ[a^2 - b^2, 0] && IntegerQ[n]

Rubi steps

\begin {align*} \int \cot ^9(c+d x) (a+a \sec (c+d x))^3 \, dx &=-\frac {a^{10} \operatorname {Subst}\left (\int \frac {x^6}{(a-a x)^5 (a+a x)^2} \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac {a^{10} \operatorname {Subst}\left (\int \left (-\frac {1}{4 a^7 (-1+x)^5}-\frac {5}{4 a^7 (-1+x)^4}-\frac {39}{16 a^7 (-1+x)^3}-\frac {9}{4 a^7 (-1+x)^2}-\frac {57}{64 a^7 (-1+x)}+\frac {1}{32 a^7 (1+x)^2}-\frac {7}{64 a^7 (1+x)}\right ) \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac {a^3}{16 d (1-\cos (c+d x))^4}+\frac {5 a^3}{12 d (1-\cos (c+d x))^3}-\frac {39 a^3}{32 d (1-\cos (c+d x))^2}+\frac {9 a^3}{4 d (1-\cos (c+d x))}+\frac {a^3}{32 d (1+\cos (c+d x))}+\frac {57 a^3 \log (1-\cos (c+d x))}{64 d}+\frac {7 a^3 \log (1+\cos (c+d x))}{64 d}\\ \end {align*}

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Mathematica [A]  time = 0.40, size = 130, normalized size = 0.87 \[ \frac {a^3 (\cos (c+d x)+1)^3 \sec ^6\left (\frac {1}{2} (c+d x)\right ) \left (-3 \csc ^8\left (\frac {1}{2} (c+d x)\right )+40 \csc ^6\left (\frac {1}{2} (c+d x)\right )-234 \csc ^4\left (\frac {1}{2} (c+d x)\right )+864 \csc ^2\left (\frac {1}{2} (c+d x)\right )+12 \left (\sec ^2\left (\frac {1}{2} (c+d x)\right )+114 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+14 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )\right )\right )}{6144 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[c + d*x]^9*(a + a*Sec[c + d*x])^3,x]

[Out]

(a^3*(1 + Cos[c + d*x])^3*Sec[(c + d*x)/2]^6*(864*Csc[(c + d*x)/2]^2 - 234*Csc[(c + d*x)/2]^4 + 40*Csc[(c + d*
x)/2]^6 - 3*Csc[(c + d*x)/2]^8 + 12*(14*Log[Cos[(c + d*x)/2]] + 114*Log[Sin[(c + d*x)/2]] + Sec[(c + d*x)/2]^2
)))/(6144*d)

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fricas [B]  time = 0.76, size = 272, normalized size = 1.83 \[ -\frac {426 \, a^{3} \cos \left (d x + c\right )^{4} - 606 \, a^{3} \cos \left (d x + c\right )^{3} - 190 \, a^{3} \cos \left (d x + c\right )^{2} + 666 \, a^{3} \cos \left (d x + c\right ) - 272 \, a^{3} - 21 \, {\left (a^{3} \cos \left (d x + c\right )^{5} - 3 \, a^{3} \cos \left (d x + c\right )^{4} + 2 \, a^{3} \cos \left (d x + c\right )^{3} + 2 \, a^{3} \cos \left (d x + c\right )^{2} - 3 \, a^{3} \cos \left (d x + c\right ) + a^{3}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 171 \, {\left (a^{3} \cos \left (d x + c\right )^{5} - 3 \, a^{3} \cos \left (d x + c\right )^{4} + 2 \, a^{3} \cos \left (d x + c\right )^{3} + 2 \, a^{3} \cos \left (d x + c\right )^{2} - 3 \, a^{3} \cos \left (d x + c\right ) + a^{3}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right )}{192 \, {\left (d \cos \left (d x + c\right )^{5} - 3 \, d \cos \left (d x + c\right )^{4} + 2 \, d \cos \left (d x + c\right )^{3} + 2 \, d \cos \left (d x + c\right )^{2} - 3 \, d \cos \left (d x + c\right ) + d\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^9*(a+a*sec(d*x+c))^3,x, algorithm="fricas")

[Out]

-1/192*(426*a^3*cos(d*x + c)^4 - 606*a^3*cos(d*x + c)^3 - 190*a^3*cos(d*x + c)^2 + 666*a^3*cos(d*x + c) - 272*
a^3 - 21*(a^3*cos(d*x + c)^5 - 3*a^3*cos(d*x + c)^4 + 2*a^3*cos(d*x + c)^3 + 2*a^3*cos(d*x + c)^2 - 3*a^3*cos(
d*x + c) + a^3)*log(1/2*cos(d*x + c) + 1/2) - 171*(a^3*cos(d*x + c)^5 - 3*a^3*cos(d*x + c)^4 + 2*a^3*cos(d*x +
c)^3 + 2*a^3*cos(d*x + c)^2 - 3*a^3*cos(d*x + c) + a^3)*log(-1/2*cos(d*x + c) + 1/2))/(d*cos(d*x + c)^5 - 3*d
*cos(d*x + c)^4 + 2*d*cos(d*x + c)^3 + 2*d*cos(d*x + c)^2 - 3*d*cos(d*x + c) + d)

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giac [A]  time = 0.51, size = 213, normalized size = 1.43 \[ \frac {684 \, a^{3} \log \left (\frac {{\left | -\cos \left (d x + c\right ) + 1 \right |}}{{\left | \cos \left (d x + c\right ) + 1 \right |}}\right ) - 768 \, a^{3} \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1 \right |}\right ) - \frac {12 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac {{\left (3 \, a^{3} + \frac {28 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {132 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {504 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {1425 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}\right )} {\left (\cos \left (d x + c\right ) + 1\right )}^{4}}{{\left (\cos \left (d x + c\right ) - 1\right )}^{4}}}{768 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^9*(a+a*sec(d*x+c))^3,x, algorithm="giac")

[Out]

1/768*(684*a^3*log(abs(-cos(d*x + c) + 1)/abs(cos(d*x + c) + 1)) - 768*a^3*log(abs(-(cos(d*x + c) - 1)/(cos(d*
x + c) + 1) + 1)) - 12*a^3*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) - (3*a^3 + 28*a^3*(cos(d*x + c) - 1)/(cos(d*x
+ c) + 1) + 132*a^3*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 + 504*a^3*(cos(d*x + c) - 1)^3/(cos(d*x + c) +
1)^3 + 1425*a^3*(cos(d*x + c) - 1)^4/(cos(d*x + c) + 1)^4)*(cos(d*x + c) + 1)^4/(cos(d*x + c) - 1)^4)/d

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maple [A]  time = 0.75, size = 141, normalized size = 0.95 \[ -\frac {a^{3} \ln \left (\sec \left (d x +c \right )\right )}{d}-\frac {a^{3}}{16 d \left (-1+\sec \left (d x +c \right )\right )^{4}}+\frac {a^{3}}{6 d \left (-1+\sec \left (d x +c \right )\right )^{3}}-\frac {11 a^{3}}{32 d \left (-1+\sec \left (d x +c \right )\right )^{2}}+\frac {13 a^{3}}{16 d \left (-1+\sec \left (d x +c \right )\right )}+\frac {57 a^{3} \ln \left (-1+\sec \left (d x +c \right )\right )}{64 d}-\frac {a^{3}}{32 d \left (1+\sec \left (d x +c \right )\right )}+\frac {7 a^{3} \ln \left (1+\sec \left (d x +c \right )\right )}{64 d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(d*x+c)^9*(a+a*sec(d*x+c))^3,x)

[Out]

-a^3/d*ln(sec(d*x+c))-1/16*a^3/d/(-1+sec(d*x+c))^4+1/6*a^3/d/(-1+sec(d*x+c))^3-11/32*a^3/d/(-1+sec(d*x+c))^2+1
3/16*a^3/d/(-1+sec(d*x+c))+57/64*a^3/d*ln(-1+sec(d*x+c))-1/32*a^3/d/(1+sec(d*x+c))+7/64*a^3/d*ln(1+sec(d*x+c))

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maxima [A]  time = 0.37, size = 142, normalized size = 0.95 \[ \frac {21 \, a^{3} \log \left (\cos \left (d x + c\right ) + 1\right ) + 171 \, a^{3} \log \left (\cos \left (d x + c\right ) - 1\right ) - \frac {2 \, {\left (213 \, a^{3} \cos \left (d x + c\right )^{4} - 303 \, a^{3} \cos \left (d x + c\right )^{3} - 95 \, a^{3} \cos \left (d x + c\right )^{2} + 333 \, a^{3} \cos \left (d x + c\right ) - 136 \, a^{3}\right )}}{\cos \left (d x + c\right )^{5} - 3 \, \cos \left (d x + c\right )^{4} + 2 \, \cos \left (d x + c\right )^{3} + 2 \, \cos \left (d x + c\right )^{2} - 3 \, \cos \left (d x + c\right ) + 1}}{192 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^9*(a+a*sec(d*x+c))^3,x, algorithm="maxima")

[Out]

1/192*(21*a^3*log(cos(d*x + c) + 1) + 171*a^3*log(cos(d*x + c) - 1) - 2*(213*a^3*cos(d*x + c)^4 - 303*a^3*cos(
d*x + c)^3 - 95*a^3*cos(d*x + c)^2 + 333*a^3*cos(d*x + c) - 136*a^3)/(cos(d*x + c)^5 - 3*cos(d*x + c)^4 + 2*co
s(d*x + c)^3 + 2*cos(d*x + c)^2 - 3*cos(d*x + c) + 1))/d

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mupad [B]  time = 1.20, size = 130, normalized size = 0.87 \[ \frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{64\,d}+\frac {57\,a^3\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{32\,d}+\frac {21\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-\frac {11\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{2}+\frac {7\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{6}-\frac {a^3}{8}}{32\,d\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}-\frac {a^3\,\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(c + d*x)^9*(a + a/cos(c + d*x))^3,x)

[Out]

(a^3*tan(c/2 + (d*x)/2)^2)/(64*d) + (57*a^3*log(tan(c/2 + (d*x)/2)))/(32*d) + ((7*a^3*tan(c/2 + (d*x)/2)^2)/6
- (11*a^3*tan(c/2 + (d*x)/2)^4)/2 + 21*a^3*tan(c/2 + (d*x)/2)^6 - a^3/8)/(32*d*tan(c/2 + (d*x)/2)^8) - (a^3*lo
g(tan(c/2 + (d*x)/2)^2 + 1))/d

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)**9*(a+a*sec(d*x+c))**3,x)

[Out]

Timed out

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